Question: The arithmetic sequence $a_i$ is defined by the formula: $a_1 = -3$ $a_i = a_{i - 1} -5$ Find the sum of the first $490$ terms in the sequence.
Getting started Let's write out the first few terms of the series: $-3 + (-8) + (-13) + (-18)...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $5$ less than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = {-3})$ and the number of terms $(n = {490})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $490 -1= 489$ terms after the first term. The sequence decreases by $5$ for each new term. So, the sequence decreases by a total of $489 \cdot 5 = 2445$ from where it starts at $-3$. That means the last term must be $-3-2445 = {-2448}$. In other words, $a_n = {-2448}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{490}}&= \dfrac {\left({-3} + ({-2448}) \right)}{2} \cdot {490} \\\\ S_{{490}} &= -1225.5 \left(490\right) \\\\ S_{{490}} &= -600{,}495\end{aligned}$ The answer $ -600{,}495 $